Question

Consider an online store where a number of customers visit and buy a product every hour. Let X be the number of people who enter the store per hour.

The store is active for 14 hours per day, every day of the week. It is calculated from data collected that the average number of customers per hour is 10.

(a) When is it appropriate to approximate a Poisson Distributed random variable with a Normal Distribution? State the appropriate parameters for the Normal Distribution.

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(b) Calculate the probability that 6 or less customers enter the store. Hint: Remember to use the correction for continuity and R to help calculate the probability.

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(c) The manager of the store decides to open up a chain of five stores and it is observed that the average number of customers entering the stores is the same. What is the mean and variance of the distribution for the average of all the five stores?

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(d) Calculate the probability for the average of the five stores when 6 or less customers enter?

Answer 1

a) If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate continuity correction is performed. The approximate Normal distribution has a mean of 10 and variance 10.

b) The probability that 6 or less customers enter the store = 0.1342.

The R code is >pnorm(6.5, 10, sqrt(10)

c) As the sum of poison distribution with the parameter nλ thus the mean=10*5 = 50 and the variance is 10*5=50 for the average sum of all the five stores. Thus for the average distribution mean is 10 and the variance is 2.

d) The probability for the average of the five stores when 6 or less customers enter is 0.00666

The R code is >pnorm(6.5, 10, sqrt(2)